On June 29th 2015, I will be traveling for the first time to Italy from USA. I will have a 10 hour layover in Moscow Russia. My question is, do I need a visa to visit the city?
The answer depends on your citizenzship. Assuming that you need a visa to ordinarily visit Russia, you would also need a visa to leave the airport. Relevant excerpt from the Sheremetyevo airport website (fairly certain you're traveling through that given your information)
В ожидании стыковочного рейса иностранные граждане могут находиться в аэропорту Шереметьево до 24 часов без российской визы. У пассажира должен быть билет авиаперевозчика на дальнейший полет с подтвержденными местами.
In English, this means
Foreign citizens waiting for a connecting flight may stay in the Sheremetyevo airport for up to 24 hours without a visa. The passenger must have a ticket from an airline for a connecting flight with confirmed seats.
When you exit your first plane at the airport, you can continue to another part of the airport (if needed) through the transit zone, which has security checks but no immigration checks. But to leave the airport, you have to go through the usual passport control desk that everyone has to use, which is where you will be prevented from exiting the airport if you need a visa to enter Russia.
Your most likely option if you want to visit Moscow is to try and get an expedited processing Russian visa, but it may be denied. You don't strictly qualify for a transit visa unless you have to change airports or your layover is more than 24 hours, and the authorities may well decide against granting you a typical tourist visa given the short duration of your planned visit. Keep in mind that Sheremetyevo is a huge airport not very close to central Moscow, and that Moscow itself is notorious for its traffic. A 10 hour layover does give you some hours in the city, but it is fairly short.